Semantical Investigations in Heyting's Intuitionistic LogicTaylor & Francis, 1981 - 287 Seiten From the point of view of non-classical logics, Heyting's implication is the smallest implication for which the deduction theorem holds. This book studies properties of logical systems having some of the classical connectives and implication in the neighbourhood of Heyt ing's implication. I have not included anything on entailment, al though it belongs to this neighbourhood, mainly because of the appearance of the Anderson-Belnap book on entailment. In the later chapters of this book, I have included material that might be of interest to the intuitionist mathematician. Originally, I intended to include more material in that spirit but I decided against it. There is no coherent body of material to include that builds naturally on the present book. There are some serious results on topological models, second order Beth and Kripke models, theories of types, etc., but it would require further research to be able to present a general theory, possibly using sheaves. That would have postponed pUblication for too long. I would like to dedicate this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from Professor Kreisel's criticism and suggestions. Professor Rabin's fun damental results on decidability and undecidability provided the powerful tools used in obtaining the majority of the results reported in this book. Professor Scott's approach to non-classical logics and especially his analysis of the Scott consequence relation makes it possible to present Heyting's logic as a beautiful, integral part of non-classical logics. |
Inhalt
PREFACE | 1 |
Introducing HPC | 20 |
The Kripke Beth and Topological | 43 |
Heytings Propositional Calculus | 63 |
CHAPTER 5 Three Intermediate Logics | 83 |
Formulas in One Variable | 108 |
Propositional Connectives | 121 |
The Interpolation Theorem | 145 |
Second Order Propositional Calculus | 159 |
Modified Kripke Interpretation | 170 |
Theories in HPC 1 | 178 |
Completeness of HPC with Respect to RE | 203 |
Undecidability Results | 227 |
Decidability Results | 266 |
Andere Ausgaben - Alle anzeigen
Semantical Investigations in Heyting's Intuitionistic Logic Dov M. Gabbay Eingeschränkte Leseprobe - 2013 |
Semantical Investigations in Heyting's Intuitionistic Logic Dov M Gabbay Keine Leseprobe verfügbar - 2014 |
Häufige Begriffe und Wortgruppen
A₁ A₂ abelian groups agrees with H An+1 Assume atomic axiom schema axioms and rules B₁ B₂ Beth structure Boston Studies C₁ Chapter Clearly complete theory connectives consistent theory constant domains construction contradiction COROLLARY deduction theorem DEFINITION denote disjunction property element equivalent Exercise exists extension of h false finite number following holds formula fragment function hence Heyting's Hilbert system homomorphism implies induction hypothesis intermediate logic intuitionistic intuitionistic logic intuitionistic propositional calculus LEMMA linear order minimal modified Kripke structure modus ponens monadic negation obtained On+1 P₁ partially ordered set Philosophy of Science predicate proof of Theorem propositional calculus propositional Kripke structure propositional variables prove quantifiers reflexive saturated and complete Scott consequence relation semantics sequence subformula substitution instance symmetric relation T₁ truth table truth value u₁ unary undecidable VxA(x
Beliebte Passagen
Seite 282 - Disjunction and existence under implication in elementary intuitionistic formalisms', JSL, 27, 11-18. Kreisel, G. and Putnam, H. 1957: 'Eine unableitungsbeweismethode fur den Intuitionistischen aussagenkalkul', Archiv f. Math. Logik, 3, 74-78. Kripke, S., 1965: 'Semantical analysis of intuitionist logic I